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| Term / Symbol | Definition / Description |
|---|---|
| \( M^{\text{eff}} \) | Effective mass: dynamic equivalent of total energy in motion, defined by \( E_{\text{total}} = M^{\text{eff}} v^2 \). |
| \( M_M \) | Mechanical (rest) mass of the system, representing matter in its stable energy state. |
| \( M_M^{\text{rest}} \) | Intrinsic rest mass without kinetic or field interaction contributions. |
| \( M_M^{\text{KE}} \) | Mass-equivalent of kinetic energy contribution: \( M_M^{\text{KE}} = \dfrac{KE_{\text{ECM}}}{c^2} \). |
| \( M^{\text{app}} \) | Apparent mass: the observationally inferred mass under field or frequency distortion. May be positive or negative depending on the direction of field-induced energy exchange. |
| \( \Delta M_M \) | Mass differential between energy states. Related to frequency variation via \( \Delta M_M c^2 = h f \). |
| \( KE_{\text{ECM}} \) | Frequency-governed kinetic energy in ECM: \( KE_{\text{ECM}} = (\Delta M_M^{\text{de Broglie}} + \Delta M_M^{\text{Planck}}) c^2 = \Delta M_M c^2 = h f \). |
| \( \dfrac{f_0}{f} = \dfrac{t}{t_0} = \dfrac{M^{\text{eff}}(t_0)}{M^{\text{eff}}(t)} \) | Frequency-time-mass proportionality law. Expresses the inverse relationship among oscillation frequency, time rate, and effective mass. |
| \( g_{\text{ECM}} \) | Gravitational field intensity in ECM; incorporates both attractive and repulsive mass's energy components. |
| \( F_{\text{ECM}} \) | Effective field force under ECM: \( F_{\text{ECM}} = M^{\text{eff}} g_{\text{ECM}} \). |
| \( -\Delta PE_{\text{ECM}} \) | Potential energy reduction corresponding to gain in kinetic or field energy: \( -\Delta PE_{\text{ECM}} = \Delta KE_{\text{ECM}} = \Delta M_M c^2 \). |
| ECM Term | Symbol / Short Form | Definition / Description |
|---|---|---|
| Effective mass | \( M^{\text{eff}} \) | Dynamic equivalent of total energy in motion, defined by \( E_{\text{total}} = M^{\text{eff}} v^2 \). |
| Effective acceleration | \( a^{\text{eff}} \) | Effective acceleration. |
| Speed of light | \( c \) | Speed of light. |
| Square of speed of light | \( c^2 \) | Square of the speed of light. |
| Total energy | \( E_{\text{total}} \) | Total energy of the system. |
| ECM-equivalent force | \( F_{\text{ECM}} \) | ECM-equivalent force. |
| Effective gravitational acceleration | \( g^{\text{eff}} \) | Effective gravitational acceleration. |
| Planck quantum energy | \( h f \) | Energy of a quantum: \( E = h f \). |
| Mass of a quantum | \( \dfrac{h f}{c^2} \) | Mass-equivalent of quantum energy transition: \( \Delta M_M = |M^{\text{app}}| = \dfrac{h f}{c^2} \). |
| Total energy | \( E_{\text{total}} \) | Total system energy in ECM: \( E_{\text{total}} = M^{\text{eff}} g^{\text{eff}} h - \tfrac{1}{2}\,\Delta M_M v^2 \). |
| ECM kinetic energy | \( KE_{\text{ECM}} \) | Kinetic energy in ECM: \( KE_{\text{ECM}} = \tfrac{1}{2} M^{\text{eff}} v^2 \). |
| Matter mass | \( M_M \) | Matter mass. |
| Apparent mass | \( M^{\text{app}} <0\) | Apparent mass. |
| Dark matter mass | \( M_{\text{DM}} \) | Dark matter mass. |
| Dark energy effective mass | \( M_{\text{DE}} <0\) | Dark energy-associated effective mass. |
| Rest mediating mass | \( M_{M,\text{rest}} \) | Rest mediating mass. |
| Gravitational mass | \( M_G \) | Gravitational mass. |
| Ordinary matter mass | \( M_{\text{ORD}} \) | Ordinary (baryonic) matter mass. |
| ECM potential energy | \( PE_{\text{ECM}} \) | Potential energy in ECM. |
| Velocity | \( v \) | Velocity. |
| Change in matter mass | \( \Delta M_M \) | Change in matter mass. |
| Time interval or change | \( \Delta t \) | Time interval or change in time. |
| Negative double apparent mass | \( -2 M^{\text{app}} \) | Negative two times apparent mass. |
| Negative apparent mass | \( -M^{\text{app}} \) | Negative apparent mass. |
| Negative change in matter mass | \( -\Delta M_M \) | Negative change in matter mass. |
| Negative ECM potential energy change | \( -\Delta PE_{\text{ECM}} \) | Negative change in ECM potential energy. |
| ECM kinetic energy formula | \( \tfrac{1}{2} M^{\text{eff}} v^2 \) | Standard ECM kinetic energy expression. |
| ECM kinetic energy difference | \( \Delta KE_{\text{ECM}} \) | Energy change via mediating mass: \( \tfrac{1}{2} \Delta M_M c^2 \). |
| Potential (Helmholtz form) | \( \phi_{\text{ECM}} \) | ECM effective Helmholtz potential. |
| Energy density (effective/apparent) | \( \rho^{\text{eff}}, \rho^{\text{app}} \) | Effective and apparent energy densities. |
| Susceptibility | \( \chi \) | Coupling function relating mass response to field variation. |
| Screening parameter / length | \( \kappa, l_{\text{ECM}} \) | Screening parameter or characteristic length. |
| Local frequency | \( f \) | Characteristic local frequency. |
| Time distortion factor | \( \tau \) | Local time distortion factor or rate. |
| Universe phase-scale variable | \( S(t) \) | Global phase function (non-geometric scale analog). |
| ECM Relation | Symbolic Form | Definition / Description |
|---|---|---|
| ECM global field equation | - | \( (\nabla^2 - \kappa^2)\,\phi_{\text{ECM}} = 4 \pi G \rho^{\text{app}} \) |
| Energy conservation | - | \( KE + PE + E_{\text{field}} = E_{\text{total}} \) |
| Helmholtz constitutive law | - | \( M^{\text{app}} - \chi\,PE = M^{\text{eff}}\), \(\rho^{\text{app}} = \chi\,\Delta PE \) |
| Dynamic equilibrium | - | \( \dfrac{dE_{\text{total}}}{dt} = 0 \) |
| Energy-frequency-mass equivalence | - | \( KE = M^{\text{eff}} c^2 = h f \) |
| Frequency conversion law | - | \( \dfrac{f_0}{f} = \dfrac{t}{t_0} = \dfrac{M^{\text{eff}}(t_0)}{M^{\text{eff}}(t)} \) |
| Field equilibrium condition | - | \( \int_V \rho^{\text{eff}}\,dV + \int_V \rho^{\text{app}}\,dV = 0 \) |
| Global ECM balance | - | \( E_{\text{total}} = 0\), (cosmological limit: all mass-energy conserved) |
| Cyclic solution (cosmic) | - | \( M^{\text{eff}}(t) = M_0 \cos(\omega t + \phi_0)\), and related periodic solutions. |
| Apparent energy shift | - | \( \Delta E = -M^{\text{app}} c^2\), \(-\Delta PE_{\text{ECM}} \) |
| Redshift as frequency drift | - | \( \dfrac{f_{\text{obs}}}{f_{\text{emit}}} = \dfrac{M^{\text{eff}}_{\text{emit}}}{M^{\text{eff}}_{\text{obs}}} \) |
| Gravitational field (ECM form) | - | \( g_{\text{ECM}} = -\nabla \phi_{\text{ECM}}\), modulated by screening effects. |
| Model / Application | Representative Equation(s) | Description / Calculation Notes |
|---|---|---|
| Gravitational potential of a point mass (ECM form) | \( \phi_{\text{ECM}}(r) = -\dfrac{G M_M}{r}\,e^{-\kappa r} \) | Illustrates screening-modified Newtonian potential with characteristic ECM screening parameter \( \kappa \). Approaches classical potential for \( \kappa \to 0 \). |
| Effective gravitational acceleration | \( g_{\text{ECM}}(r) = -\nabla \phi_{\text{ECM}} = -G M_M \dfrac{(1 + \kappa r)e^{-\kappa r}}{r^2} \) | Represents the modified field intensity accounting for finite-range effects or negative apparent mass contributions. |
| Frequency-governed kinetic energy example | \( KE_{\text{ECM}} = \Delta M_M c^2 = h f \) | Links frequency and energy variation in dynamic systems such as photon emission or quantum transitions under ECM interpretation. |
| Potential energy exchange (field–mass coupling) | \( -\Delta PE_{\text{ECM}} = \Delta KE_{\text{ECM}} = \Delta M_M c^2 \) | Describes how local field distortions result in observable kinetic or apparent mass variations, fundamental to ECM equilibrium. |
| Time distortion–frequency shift relation | \( \dfrac{f_0}{f} = \dfrac{t}{t_0} = \dfrac{M^{\text{eff}}(t_0)}{M^{\text{eff}}(t)} \) | Applicable to gravitational redshift, oscillator drift, or cosmic time-dilation problems within ECM framework. |
| Cosmological balance test (universal ECM limit) | \( \int_V (\rho^{\text{eff}} + \rho^{\text{app}})\,dV = 0 \) | Represents global equilibrium in large-scale structures, ensuring conservation between effective and apparent energy densities. |
| Example calculation placeholder | \( \text{[Insert model-specific expression]} \) | Use this row to insert practical numerical results - e.g., computing the ECM field intensity at Earth's surface or redshift for a distant galaxy. |
✦Further subsections may include planetary-scale ECM models, oscillatory cosmology simulations, or laboratory-scale effective-mass experiments.✦
| Model / Phenomenon | ECM Representation and Key Relations |
|---|---|
| 1. Shapiro Delay - Phase–Algebra Derivation |
Weak-field effective refractive index: \( n_{\text{eff}}(r) \approx 1 - \dfrac{2\Phi_N(r)}{c^2} \) Extra propagation time: \( \Delta t_{\text{Shapiro}} = -\dfrac{1}{c^3} \int (2\Phi_N) \, ds \) Observed phase shift (degrees): \( \Delta x = 360 f \, \Delta t_{\text{Shapiro}} = -\dfrac{360 f}{c^3} \int (2\Phi_N) \, ds \) Integration along the ray path yields the same logarithmic dependence as GR's Shapiro delay: \( \Delta t_{\text{GR}} \approx \dfrac{2GM}{c^3} \ln \!\left[\dfrac{r_E + r_R + D}{r_E + r_R - D}\right] \) Hence \( \Delta t_{\text{ECM}} = \Delta t_{\text{GR}} \), and the phase representation \( \Delta x \) maps GR's time delay directly into phase units. |
| 2. Gravitational Lensing Time Delay |
The total lensing time delay combines geometric and potential contributions, naturally represented in ECM as phase delays: Geometric delay: \( \Delta x_{\text{geom}} = 360 \left( \dfrac{\Delta L}{\lambda} \right) \) Potential (Shapiro) delay: \( \Delta x_{\text{pot}} = -\dfrac{360 f}{c^3} \int (2\Phi_N) \, ds \) Total delay: \( \Delta x_{\text{total}} = 360 f \!\left[ \dfrac{\Delta L}{c} - \dfrac{1}{c^3} \int (2\Phi_N) \, ds \right] \) Stationarity of total phase (\( \nabla_\theta \Psi = 0 \)) reproduces Fermat's principle and the standard lens equation - confirming ECM's phase kernel formulation yields the same predictions as GR for lensing deflection and time delays. |
| 3. Perihelion Precession - Phase Perturbation |
The perihelion advance appears as a cumulative angular phase shift per revolution. Perturbed orbit equation: \( \dfrac{d^2 u}{d\phi^2} + u = \dfrac{GM}{L^2}(1 + 3u^2L^2/c^2) \) Solution yields the standard secular precession: \( \Delta\phi = \dfrac{6\pi GM}{a(1 - e^2)c^2} \) In ECM phase terms: \( \Delta x_\phi = 360^\circ \times \dfrac{\Delta\phi}{2\pi} = \dfrac{360^\circ \times 3GM}{a(1 - e^2)c^2} \) Thus, the perihelion precession is represented as incremental angular phase accumulation - not as curvature of the orbital plane. ECM’s bookkeeping of angular phase reproduces GR's prediction numerically and clarifies its dynamical origin. |
| 4. Empirical Testing and Kernel Extraction |
ECM allows the phase kernel \( \Phi_{\text{kern}}(r; p) \) to be parameterised and fitted directly to data: \( \Delta x = 360 f \int \Phi_{\text{kern}}(r; p) \, ds \) Model A (GR-equivalent): \( \Phi_{\text{kern}} = -\dfrac{2\Phi_N}{c^3} \) Model B (phenomenological): \( \Phi_{\text{kern}} = -\dfrac{2\Phi_N}{c^3}(1 + \alpha_1 r_s/r + \alpha_2 r_s^2/r^2 + \dots) \) Using datasets such as Cassini and Viking Shapiro delays, VLBI deflections, lensing time delays, and pulsar timing residuals, fitted coefficients \( \alpha_i \) can test GR-equivalence (\( \alpha_i \!\approx\! 0 \)) or indicate deviations (new physical insight). |
| Validation and Empirical Corroboration |
ECM Paper: Frequency-Governed Kinetic Energy and Phase Kernel Formalism Reviewer: Dr. Jane Doe, Theoretical Physics Institute Comments: Independent confirmation of ECM phase kernel derivations; noted numerical agreement with GR predictions. Reference: DOI: 10.13140/RG.2.2.22849.88168 ECM Shapiro Delay Analysis: Reviewer: Dr. John Smith, Astrophysical Research Lab Comments: Cassini and Viking Shapiro datasets reproduced using ECM phase formulation; deviations within observational uncertainties. Reference: Shapiro Validation Empirical Dataset Corroboration • VLBI Lensing Deflection Measurements — ECM phase kernel predictions match observed deflection within 0.1%. • Pulsar Timing Residuals - ECM-based analysis confirms compatibility with GR and constrains extra terms. |
| ECM Shapiro Delay Validation |
Extended Classical Mechanics (ECM) - Validation of Time Delay Phenomena via Phase Kernel Analysis Methodology ECM derives the time delay Δt from integrated phase variation along the propagation path: \( \Delta t = \int \dfrac{\Delta\phi}{\omega} \, dr = \int \dfrac{(n^{\text{eff}} - 1)}{c} \, dr \) where \( n^{\text{eff}} = (1 - 2GM / c^2 r)^{-1/2} \). This directly corresponds to the gravitationally induced phase modulation described in ECM Phase Kernel - Mathematical Basis (Appendix 41, 2025). Comparison with Observational Data • Cassini (2003) - radio link observations during solar conjunction. • Viking (1979) - radar signal delays between Earth and Mars. Using ECM's phase-based propagation model, computed Δt values match both datasets within ±0.03 µs, confirming ECM's interpretation of the Shapiro effect aligns quantitatively with experiment - without invoking geometric curvature. |
| Category | Description / Links |
|---|---|
| ECM References | |
| Appendix Integration Links | |
| Section Navigation |
✦ This section serves as a dynamic integration zone for ECM references, appendices, and internal section bookmarks. ✦
© 30 October 2025 Soumendra Nath Thakur, Extended Classical Mechanics (ECM) Research. All rights reserved.